Non-Uniqueness in Plane Fluid Flows

TL;DR

This paper explores non-uniqueness in plane fluid flows, demonstrating how multiple solutions can exist for the same initial conditions, and shows how a maximal entropy rate criterion can restore uniqueness.

Contribution

It interprets non-unique solutions as non-unique Lagrange trajectories and links these to steady compressible fluid flows satisfying Euler and Navier-Stokes equations.

Findings

Non-unique solutions can be characterized as Lagrange trajectories.

Maximal entropy rate criterion restores solution uniqueness.

Solutions are consistent with steady compressible fluid flows.

Abstract

Examples of dynamical systems proposed by Z. Artstein and C. M. Dafermos admit non-unique solutions that track a one parameter family of closed circular orbits contiguous at a single point. Switching between orbits at this single point produces an infinite number of solutions with the same initial data. Dafermos appeals to a maximal entropy rate criterion to recover uniqueness. These results are here interpreted as non-unique Lagrange trajectories on a particular spatial region. The corresponding velocity is proved consistent with plane steady compressible fluid flows that for specified pressure and mass density satisfy not only the Euler equations but also the Navier-Stokes equations for specially chosen volume and (positive) shear viscosities. The maximal entropy rate criterion recovers uniqueness.